Euler (1735)

Abstract

Daniel Bernoulli wrote Euler on 18 December 1734 that he was studying the small (transversal) vibrations of a rod with one end fixed in a wall.1 On 4 May 1735, he wrote that he had found the equation for its shape, namely â d4 y/ dz4 = y, but that its solutions known to him, namely sine and exponential functions, were inappropriate.2 Euler responded that he also had found the equation but only the series form of the solution.3 In October of 1735, Euler presented the St. Petersburg Academy with a paper in which he derived the equation, dealt with the boundary conditions and gave the fundamental solution in series form.4 This paper is of further significance, however, because in it Euler presented a lucid overview of vibration theory as he understood it at the time. He gave the first explicit discussion of the pendulum condition which, in turn, he looked on as a static equilibrium condition. This reduced dynamical (vibration) problems to static problems which were understood with more sophistication. The paper appeared in 1740.